Definition df-log 25140

Description: Define the natural logarithm function on complex numbers. It is defined as the principal value, that is, the inverse of the exponential whose imaginary part lies in the interval (-pi, pi]. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Paul Chapman, 21-Apr-2008.)

Assertion

Ref	Expression
df-log	⊢ log = ◡(exp ↾ (◡ℑ “ (-π(,]π)))

Detailed syntax breakdown of Definition df-log

Step	Hyp	Ref	Expression
1		clog 25138	. 2 class log
2		ce 15415	. . . 4 class exp
3		cim 14457	. . . . . 6 class ℑ
4	3	ccnv 5554	. . . . 5 class ◡ℑ
5		cpi 15420	. . . . . . 7 class π
6	5	cneg 10871	. . . . . 6 class -π
7		cioc 12740	. . . . . 6 class (,]
8	6, 5, 7	co 7156	. . . . 5 class (-π(,]π)
9	4, 8	cima 5558	. . . 4 class (◡ℑ “ (-π(,]π))
10	2, 9	cres 5557	. . 3 class (exp ↾ (◡ℑ “ (-π(,]π)))
11	10	ccnv 5554	. 2 class ◡(exp ↾ (◡ℑ “ (-π(,]π)))
12	1, 11	wceq 1537	1 wff log = ◡(exp ↾ (◡ℑ “ (-π(,]π)))

This definition is referenced by:  logrn  25142  dflog2  25144  dvlog  25234  efopnlem2  25240